...-iteration

The research of the first two authors, was partially supported by the Research Council K.U.Leuven, project OT/05/40 (Large rank structured matrix computations), CoE EF/05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research-Flanders (Belgium), G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), G.0423.05 (RAM: Rational modelling: optimal conditioning and stable algorithms), and by the Interuniversity Poles of Attraction Poles, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The first author has a grant as ``Postdoctoraal Onderzoeker'' from the Fund for Scientific Research-Flanders (Belgium). The work of the third author was partially supported by MIUR, grant number 2004015437. The scientific responsibility rests with the authors.

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...-algorithms

This covers a more general class of iterations such as $QR$-methods, $LR$-methods and so forth.

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... etc.

Even though not mentioned, in case of a band matrix, a tridiagonal matrix, ... we implicitly assume the matrices to be symmetric.

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....

With essentially equivalent is meant that the elements are the same up to a factor, where all factors are roots of unity.

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...#tex2html_wrap_inline3752#

In some cases one forces convergence to blocks on the diagonal, which then contain the eigenvalues.

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...$simple$

A matrix is called simple if it has $n$ linearly independent eigenvectors.

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...$spectral$

The spectral norm is naturally induced by the $||.||_2$ norm on vectors.

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... transformations

We remark that other types of factorizations based on Gauss transforms can also be computed in a similar fashion.

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... element.

Similarly one can choose the Wilkinson shift. Here we used the Rayleigh shift.

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